Upon his retirement in 1990 as president of the ICMI (International Commission on Mathematical Instruction), Jean-Pierre Kahane spoke perceptively of the intimate connection between mathematics and mathematics education in the following terms:

In no other living science is the part of presentation, of the transformation of disciplinary knowledge to knowledge as it is to be taught (transformation didactique) so important at a research level.

In no other discipline, however, is the distance between the taught and the new so large.

In no other science has teaching and learning such social importance.

In no other science is there such an old tradition of scientists’ commitment to educational questions.

The recent discussion of Calculus in O notation gave a wonderful proof how close the concept of didactic transformation is to the hearts of mathematicians/computer scientists—even if the the words themselves are not in common use. I thank all contributors for taking part in the discussion.

Later in April I will give a talk on Didactic Transformation at a curious meeting of British HE educationalists, titled The Teaching – Research Interface: Implications for Practice in HE and FE. I am setting myself an unrealistic aim to try to find arguments in support of a simple thesis: teaching of mathematics is very different from teaching any other discipline and for that reason mathematics should be treated differently from other university disciplines.

The concept of didactic transformation is the principal stumbling block; I know from experience how difficult it is to sell to educationalists the idea that didactic transformation of mathematical material is, first of and above all, a mathematical problem.

A brief historic note: the concept of transformation didactique can be traced back to Auguste Comte.

A discourse, then, which is in the full sense didactic, ought to differ essentially from one simply logical, in which the thinker freely follow his own course, paying no attention to the natural conditions of all communication. […]

On the other hand, this transformation for the purposes of teaching is only practicable where the doctrines are sufficiently worked out for us to be able to distinctly compare the different methods of expanding them as a whole and to easily foresee the objections which they will naturally elicit.